When a gas is heated under constant pressure conditions, the heat is required for raising the temperature of the gas and also for doing mechanical work against the external pressure during expansion.
Consider one mole of gas to expand from volume V1 to V2 at a constant pressure p, when the temperature is raised from T K to (T + 1) K. Then by definition
CP - CV = work done by the gas due to expansion. Work done by the gas during expansion
= pΔV = p(V2 - V1)
Therefore, CP - CV = p(V2 - V1)
For one mole of the gas, using gas equation, one can write
pV2 = R (T + 1)
and pV1 = RT
or, pV2 - pV1 = R(T + 1) - RT
= R(T + 1) - RT = R
or, p(V2 - V1) = R
Therefore, CP - CV = R
